in antiquity, a donative or lar- Calcium gels bestowed on Roman soldiers for buying shoes. In monasteries, *calcarium* denoted the daily service of cleaning the shoes of the religious.
**CALCEOLARIA.** See Botany Index.
**CALCHAS,** in fabulous history, a famous diviner, followed the Greek army to Troy. He foretold that the siege would last ten years; and that the fleet, which was detained in the port of Aulis by contrary winds, would not fail till Agamemnon's daughter had been sacrificed to Diana. After the taking of Troy, he retired to Colophon; where, it is said, he died of grief, because he could not divine what another of his profession, called *Mopfus,* had discovered.
**CALCINATION,** in Chemistry, the reducing of substances to a calx, or powder, by fire. Limestone is said to be calcined or burned by being deprived of its carbonic acid, and thus brought to the caustic state. But when a metallic substance is calcined by being exposed to strong heat, it assumes the form of powder or calx, by combining with oxygen. See Chemistry Index.
**CALCINATO,** a town of Italy, in the duchy of Mantua, remarkable for a victory gained over the Imperialists by the French in 1706. E. Long. 9. 55. N. Lat. 45. 25.
**CALCULARY of a Pear,** a congeries of little strong knots dispersed through the whole parenchyma of the fruit. The calculary is most observed in rough-tufted or choke pears. The knots lie more continuous and compact together towards the pear where they surround the acetary. About the stalk they stand more dilatant; but towards the cork, or stool of the flower, they still grow closer, and there at last gather into the firmness of a plum stone. The calculary is no vital or essential part of the fruit; the several knots whereof it consists being only to many concretions or precipitations out of the sap, as we see in urines, wines, and other liquors.
**CALCULATION,** the act of computing several sums, by adding, subtracting, multiplying, or dividing. See Arithmetic.
Calculation is more particularly used to signify the computations in astronomy and geometry, for making tables of logarithms, ephemerides, finding the time of eclipses, &c. See Astronomy, Geometry, and Logarithms.
**CALCULUS,** primarily denotes a little stone or pebble, anciently used in making computations, taking of suffrages, playing at tables, and the like. In after times, pieces of ivory, and counters struck of silver, gold, and other matters, were used in lieu thereof, but still retaining the ancient names. Computists were by the lawyers called *calculones,* when they were either slaves, or newly freed men; those of a better condition were named *calculatores* or *numerarii:* ordinarily there was one of these in each family of distinction. The Roman judges anciently gave their opinions by calculi, which were white for abolution, and black for condemnation. Hence *calculus albus,* in ancient writers, denotes a favourable vote, either in a person to be absolved and acquitted of a charge, or elected to some dignity or post; as *calculus niger* did the contrary. This usage is said to have been borrowed from the Thracians, who marked their happy or prosperous days by *white,* and their unhappy by *black,* pebbles, put each night into an urn.
Besides the diversity of colour, there were some calculi also which had figures or characters engraven on them, as those which were in use in taking the suffrages both in the senate and at assemblies of the people. These calculi were made of thin wood, polished and covered over with wax. Their form is still seen in some medals of the Caecilian family; and the manner of casting them into the urns, in the medals of the Licinian family. The letters marked upon these calculi were U. R. for *uti rogar,* and A. for *antiquo,* the first of which expressed an approbation of the law, the latter a rejection of it. Afterwards the judges who sat in capital causes used calculi marked with the letter A., for *abrogare,* C. for *condemno,* and N. L. for *non ligner,* signifying that a more full information was required.
**CALCULUS** is also used in ancient geometric writers for a kind of weight equal to two grains of cicer. Some make it equivalent to the libra, which is equal to three grains of barley. Two calculi made the centurum.
**CALCULUS,** in Mathematics, is a certain method of performing investigations and resolutions, particularly in mechanical philosophy. Thus there is the Differential calculus, the Exponential, the Integral, the Literal, and the Antecedental.
**CALCULUS Differentialis,** is a method of differencing quantities, or of finding an infinitely small quantity, which being taken infinite times, shall be equal to a given quantity; or, it is the arithmetic of the infinitely small differences of variable quantities.
The foundation of this calculus is an infinitely small quantity, or an infinitesimal, which is a portion of a quantity incomparable to that quantity, or that is less than any assignable one, and therefore accounted as nothing; the error accruing by omitting it being less than any assignable one. Hence two quantities, only differing by an infinitesimal, are reputed equal. Thus, in astronomy, the diameter of the earth is an infinitesimal, in respect of the distance of the fixed stars; and the same holds in abstract quantities. The term, infinitesimal, therefore, is merely relative, and involves a relation to another quantity; and does not denote any real ens or being. Now infinitesimals are called differentials, or differential quantities, when they are considered as the differences of two quantities. Sir Isaac Newton calls them moments; considering them as the momentary increments of quantities, v. g. of a line generated by the flux of a point, or of a surface by the flux of a line. The differential calculus, therefore, and the doctrine of fluxions, are the same thing under different names; the former given by M. Leibnitz, and the latter by Sir Isaac Newton: each of whom lays claim to the discovery. There is, indeed, a difference in the manner of expressing the quantities resulting from the different views wherein the two authors consider the infinitesimals: the one as moments, the other as differences. Leibnitz, and most foreigners, express the differentials of quantities by the same letters as variable ones, only prefixing the letter $d$: thus the differential of $x$ is called $dx$; and that of $y$, $dy$: now $dx$ is a positive quantity, if $x$ continually increase; negative, if it decrease. The English, with Sir Isaac Newton, Newton, instead of \( dx \) write \( x \) (with a dot over it), for \( dy \), \( y \), &c. which foreigners object against, on account of that confusion of points, which they imagine arises when differentials are again differenced; besides, that the printers are more apt to overlook a point than a letter. Stable quantities being always expressed by the first letters of the alphabet \( da = o \), \( db = o \), \( dc = o \); therefore \( d(x+y-a) = dx + dy \), and \( d(x-y+a) = dx - dy \). So that the differencing of quantities is easily performed by the addition or subtraction of their compounds.
To difference quantities that multiply each other; the rule is, first, multiply the differential of one factor into the other factor, the sum of the two factors is the differential sought: thus, the quantities being \( x, y \), the differential will be \( xdy + ydx \), i.e. \( d(xy) = xdy + ydx \). Secondly, If there be three quantities mutually multiplying each other, the factum of the two must then be multiplied into the differential of the third; thus suppose \( vxy \), let \( v = t \), then \( vx = ty \); consequently \( d(vxy) = tdxy + ydv \); but \( dt = vdx + xdv \). These values, therefore, being substituted in the antecedent differential, \( tdy + ydv \), the result is, \( d(vxy) = vxdy + ydx + xydv \). Hence it is easy to apprehend how to proceed, where the quantities are more than three. If one variable quantity increase, while the other decreases, it is evident \( ydx - xdy \) will be the differential of \( xy \).
To difference quantities that mutually divide each other; the rule is, first, multiply the differential of the divisor into the dividend; and on the contrary, the differential of the dividend into the divisor; subtract the last product from the first, and divide the remainder by the square of the divisor, the quotient is the differential of the quantities mutually dividing each other. See Fluxions.
**Calculus Exponentialis**, is a method of differencing exponential quantities, or of finding and summing up the differentials or moments of exponential quantities; or at least bringing them to geometrical constructions.
By exponential quantity, is here understood a power, whose exponent is variable; v.g., \( x^x \), \( e^x \), \( x^y \), where the exponent \( x \) does not denote the same in all the points of a curve, but in some stands for 2, in others for 3, in others for 5, &c.
To difference an exponential quantity; there is nothing required but to reduce the exponential quantities to logarithmic ones; which done, the differencing is managed as in logarithmic quantities. Thus, suppose the differential of the exponential quantity \( x^y \) required, let
\[ x^y = z \]
Then will \( ylzx = lzx \)
\[ lxdy + \frac{ydx}{x} = dz \]
\[ z lxdy + \frac{x^y dx}{x} = dz \]
That is, \( x^y lxdy + x^y dx = dz \).
**Calculus Integralis**, or **Summatorius**, is a method of integrating, or summing up moments, or differential quantities; i.e. from a differential quantity given, to find the quantity from whose differencing the given differential results.
The integral calculus, therefore, is the inverse of the differential one: whence the English, who usually call the differential method fluxions, give this calculus, which ascends from the fluxions, to the flowing or variable quantities; or as foreigners express it, from the differences to the sums, by the name of the inverse method of fluxions.
Hence, the integration is known to be justly performed, if the quantity found, according to the rules of the differential calculus, being differenced, produce that proposed to be summed.
Suppose \( f \) the sign of the sum, or integral quantity, then \( \int f dx \) will denote the sum, or integral of the differential \( ydx \).
To integrate, or sum up a differential quantity: it is demonstrated, first, that \( f dx = x \): secondly, \( f(dx + dy) = x + y \): thirdly, \( f(xdy + ydx) = xy \): fourthly, \( f(m^n - 1dx) = m^n - m \): fifthly, \( f(n:m) = n:m \): sixthly, \( f(ydx - xdy) : y^2 = x : y \). Of these, the fourth and fifth cases are the most frequent, wherein the differential quantity is integrated, by adding a variable unity to the exponent, and dividing the sum by the new exponent multiplied into the differential of the root; v.g., the fourth case, by \( m - (1 + 1)dx \), i.e. by \( m^2dx \).
If the differential quantity to be integrated doth not come under any of these formulas, it must either be reduced to an integral finite, or an infinite series, each of whose terms may be summed.
It may be here observed, that, as in the analysis of finites, any quantity may be raised to any degree of power; but vice versa, the root cannot be extracted out of any number required; so in the analysis of infinites, any variable or flowing quantity may be differenced; but vice versa, any differential cannot be integrated. And as, in the analysis of finites, we are not yet arrived at a method of extracting the roots of all equations, so neither has the integral calculus arrived at its perfection: and as in the former we are obliged to have recourse to approximation, so in the latter we have recourse to infinite series, where we cannot attain to a perfect integration.
**Calculus Lateralis**, or **Literal Calculus**, is the same with spurious arithmetic, or algebra, so called from its using the letters of the alphabet; in contradiction to numeral arithmetic, which uses figures. In the literal calculus given quantities are expressed by the first letters, \( a, b, c, d \); and quantities sought by the last \( x, y, z \), &c. Equal quantities are denoted by the same letters.
**Calculus Antecedental**, a geometrical method of reasoning invented by Mr Glenie, which, without any consideration of motion or velocity, is applicable to all the purposes of fluxions. In this method, says Mr Glenie, "every expression is truly and strictly geometrical, is founded on principles frequently made use of by the ancient geometers, principles admitted into the very first elements of geometry, and repeatedly used by Euclid himself. As it is a branch of general geometrical proportion, or universal comparison, and is derived from an examination of the antecedents of ratios, hav- Calculus. ing given consequents and a given standard of comparison in various degrees of augmentation and diminution they undergo by composition and decomposition, I have called it the antecedent calculus. As it is purely geometrical, and perfectly scientific, I have, since it first occurred to me in 1779, always made use of it instead of the fluxionary and differential calculi, which are merely arithmetical. Its principles are totally unconnected with the ideas of motion and time, which, strictly speaking, are foreign to pure geometry and abstract science, though, in mixed mathematics and natural philosophy, they are equally applicable to every investigation, involving the consideration of either with the two numerical methods just mentioned. And as many such investigations require compositions and decompositions of ratios, extending greatly beyond the triplicate and subtriplicate, this calculus in all of them furnishes every expression in a strictly geometrical form.
The standards of comparison in it may be any magnitudes whatever, and are of course indefinite and innumerable; and the consequents of the ratios, compound or decomposed, may be either equal or unequal, homogeneous or heterogeneous. In the fluxionary and differential methods, on the other hand, 1, or unit, is not only the standard of comparison, but also the consequent of every ratio compounded or decomposed." See Phil. Trans. Edin. vol. iv.
Some mathematicians, however, are of opinion that the advantage to be derived from the employment of this calculus is not so great as the author seems to promise from it.
Calculus Minerve, among the ancient lawyers, denoted the decision of a cause, wherein the judges were equally divided. The expression is taken from the history of Orestes, represented by Aeschylus and Euripides; at whose trial, before the Areopagites, for the murder of his mother, the votes being equally divided for and against him, Minerva interposed, and gave the casting vote or calculus in his behalf.
M. Cramer, professor at Marburg, has a discourse expressed, De Calculo Minerve; wherein he maintains, that all the effect an entire equality of voices can have, is to leave the cause in statu quo.
Calculus Tiburtinus, a sort of figured stone, formed in great plenty about the catacarts of the Anio, and other rivers in Italy; of a white colour, and in shape oblong, round, or echinated. They are a species of the striae lapidea, or flabellites, and generated like them; and fo like sugar plums, that it is a common jest at Rome to deceive the unexperienced by serving them up as deferts.